Automorphism groups of superextensions of groups

T. O. Banakh; V. M. Gavrylkiv

doi:10.15330/ms.48.1.134-142

A family $\mathcal L$ of subsets of a set $X$ is called {\em linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked family $\mathcal M$ is {\em maximal linked} if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $\lambda(X)$ consists of all maximal linked families on $X$. Any associative binary operation $:X\times X \to X$ can be extended to an associative binary operation $: \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of order $\leq 5$.

1. T. Banakh, V. Gavrylkiv, Algebra in superextension of groups, II: cancelativity and centers, Algebra Discrete Math., 4 (2008), 1–14.

2. T. Banakh, V. Gavrylkiv, Algebra in superextension of groups: minimal left ideals, Mat. Stud., 31 (2009), №2, 142–148.

3. T. Banakh, V. Gavrylkiv, Extending binary operations to functor-spaces, Carpathian Math. Publ., 1 (2009), №2, 113–126.

4. T. Banakh, V. Gavrylkiv, Algebra in the superextensions of twinic groups, Dissertationes Math., 473 (2010), 3-74.

5. T. Banakh, V. Gavrylkiv, Algebra in superextensions of semilattices, Algebra Discrete Math., 13 (2012), №1, 26-42.

6. T. Banakh, V. Gavrylkiv, Algebra in superextensions of inverse semigroups, Algebra Discrete Math., 13 (2012), №2, 147-168.

7. T. Banakh, V. Gavrylkiv, Characterizing semigroups with commutative superextensions, Algebra Discrete Math., 17 (2014), №2, 161-192.

8. T. Banakh, V. Gavrylkiv, On structure of the semigroups of k-linked upfamilies on groups, Asian- European J. Math., 10 (2017), №4, 15 p.

9. T. Banakh, V. Gavrylkiv, O. Nykyforchyn, Algebra in superextensions of groups, I: zeros and commutativity, Algebra Discrete Math., 3 (2008), 1-29.

10. A.E. Brouwer, C.F. Mills, W.H. Mills, A. Verbeek, Counting families of mutually intesecting sets, Electron. J. Combin., 20 (2013), №2, 8 p.

11. V. Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces, Mat. Stud., 28 (2007), №1, 92-110.

12. V. Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces, Mat. Stud., 29 (2008), №1, 18-34.

13. V. Gavrylkiv, On representation of semigroups of inclusion hyperspaces, Carpathian Math. Publ., 2 (2010), №1, 24-34.

14. V. Gavrylkiv, Superextensions of cyclic semigroups, Carpathian Math. Publ., 5 (2013), №1, 36-43.

15. V. Gavrylkiv, Semigroups of centered upfamilies on finite monogenic semigroups, J. Algebra, Number Theory: Adv. App., 16 (2016), №2, 71-84.

16. V. Gavrylkiv, Semigroups of centered upfamilies on groups, Lobachevskii J. Math., 38 (2017), №3, 420-428.

17. V. Gavrylkiv, Superextensions of three-element semigroups, Carpathian Math. Publ., 9 (2017), №1, 28-36.

18. V. Gavrylkiv, On the automorphism group of the superextension of a semigroup, Mat. Stud., 48 (2017), №1, 3-13.

19. N. Hindman, D. Strauss, Algebra in the Stone-.Cech compactification, de Gruyter, Berlin, New York, 1998.

20. J.M. Howie, Fundamentals of semigroup theory, The Clarendon Press Oxford University Press, New York, 1995.

21. J. van Mill, Supercompactness and Wallman spaces, Mathematical Centre Tracts, V.85 Amsterdam, 1977.

22. D. Robinson, A course in the theory of groups, Graduate Texts in Mathematics, V.80 (Springer-Verlag, New York, 1996).

23. A. Teleiko, M. Zarichnyi, Categorical Topology of Compact Hausdofff Spaces, V.5, VNTL, Lviv, 1999.

24. A. Verbeek, Superextensions of topological spaces, Mathematical Centre Tracts, V.41, Amsterdam, 1972.

	Automorphism groups of superextensions of groups
Author	T. O. Banakh, V. M. Gavrylkiv t.o.banakh@gmail.com; vgavrylkiv@gmail.com Ivan Franko National University of Lviv, Ukraine, and Institute of Mathematics, Jan Kochanowski University in Kielce, Poland; Faculty of Mathematics and Computer Science Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Abstract	A family $\mathcal L$ of subsets of a set $X$ is called {\em linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked family $\mathcal M$ is {\em maximal linked} if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $\lambda(X)$ consists of all maximal linked families on $X$. Any associative binary operation $:X\times X \to X$ can be extended to an associative binary operation $: \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of order $\leq 5$.
Keywords	group; semigroup; maximal linked family; superextension; automorphism group
DOI	doi:10.15330/ms.48.2.134-142
Reference	1. T. Banakh, V. Gavrylkiv, Algebra in superextension of groups, II: cancelativity and centers, Algebra Discrete Math., 4 (2008), 1–14. 2. T. Banakh, V. Gavrylkiv, Algebra in superextension of groups: minimal left ideals, Mat. Stud., 31 (2009), №2, 142–148. 3. T. Banakh, V. Gavrylkiv, Extending binary operations to functor-spaces, Carpathian Math. Publ., 1 (2009), №2, 113–126. 4. T. Banakh, V. Gavrylkiv, Algebra in the superextensions of twinic groups, Dissertationes Math., 473 (2010), 3-74. 5. T. Banakh, V. Gavrylkiv, Algebra in superextensions of semilattices, Algebra Discrete Math., 13 (2012), №1, 26-42. 6. T. Banakh, V. Gavrylkiv, Algebra in superextensions of inverse semigroups, Algebra Discrete Math., 13 (2012), №2, 147-168. 7. T. Banakh, V. Gavrylkiv, Characterizing semigroups with commutative superextensions, Algebra Discrete Math., 17 (2014), №2, 161-192. 8. T. Banakh, V. Gavrylkiv, On structure of the semigroups of k-linked upfamilies on groups, Asian- European J. Math., 10 (2017), №4, 15 p. 9. T. Banakh, V. Gavrylkiv, O. Nykyforchyn, Algebra in superextensions of groups, I: zeros and commutativity, Algebra Discrete Math., 3 (2008), 1-29. 10. A.E. Brouwer, C.F. Mills, W.H. Mills, A. Verbeek, Counting families of mutually intesecting sets, Electron. J. Combin., 20 (2013), №2, 8 p. 11. V. Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces, Mat. Stud., 28 (2007), №1, 92-110. 12. V. Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces, Mat. Stud., 29 (2008), №1, 18-34. 13. V. Gavrylkiv, On representation of semigroups of inclusion hyperspaces, Carpathian Math. Publ., 2 (2010), №1, 24-34. 14. V. Gavrylkiv, Superextensions of cyclic semigroups, Carpathian Math. Publ., 5 (2013), №1, 36-43. 15. V. Gavrylkiv, Semigroups of centered upfamilies on finite monogenic semigroups, J. Algebra, Number Theory: Adv. App., 16 (2016), №2, 71-84. 16. V. Gavrylkiv, Semigroups of centered upfamilies on groups, Lobachevskii J. Math., 38 (2017), №3, 420-428. 17. V. Gavrylkiv, Superextensions of three-element semigroups, Carpathian Math. Publ., 9 (2017), №1, 28-36. 18. V. Gavrylkiv, On the automorphism group of the superextension of a semigroup, Mat. Stud., 48 (2017), №1, 3-13. 19. N. Hindman, D. Strauss, Algebra in the Stone-.Cech compactification, de Gruyter, Berlin, New York, 1998. 20. J.M. Howie, Fundamentals of semigroup theory, The Clarendon Press Oxford University Press, New York, 1995. 21. J. van Mill, Supercompactness and Wallman spaces, Mathematical Centre Tracts, V.85 Amsterdam, 1977. 22. D. Robinson, A course in the theory of groups, Graduate Texts in Mathematics, V.80 (Springer-Verlag, New York, 1996). 23. A. Teleiko, M. Zarichnyi, Categorical Topology of Compact Hausdofff Spaces, V.5, VNTL, Lviv, 1999. 24. A. Verbeek, Superextensions of topological spaces, Mathematical Centre Tracts, V.41, Amsterdam, 1972.
Pages	134-142
Volume	48
Issue	2
Year	2017
Journal	Matematychni Studii
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Table of content of issue	html